The resistors are called pull-down resistors, and are intended to keep the inputs to the NAND gates low until the button is pressed. My guess is that you have the resistors on the wrong side of the switches. Check your assembly again to be sure that the resistors go directly from ground to the NAND gate inputs and not through the switches.

your resistors need to go to supply and the input of your nand and the switch needs to pull the input of the gate low in order to light your led with the switch closed ....just as an example if you were using a 4093,sc pin 1 and 2,10k resistor from 1 and 2 to supply ,switch between 1 and 2 to ground and pin 3 will swing hi with the switch operated and low with the switch open.....

the difference you have is that the original circuit uses 74HCxx series chip, while you are using the really old 74xx series. These chips require quite substantial current to be pulled from the inputs to register it as a low level, according to datasheet this current is up to 1.6mA. This is why it works correctly with 1.5k pulldowns but not with 10k.

I dont think I would have guessed it without the photos, the original TTL (non-cmos) chips are quite rare nowadays, and I doubt they are still being manufactured. Normally you would use a HC or HCT versions, as these are the most commont that you can buy.

Ok I checked on farnell and a 7400 costs about three times more than 74HC00, so I guess you can still get them, or at least some from that series, but I think they will be new old stock and not manufactured anymore.